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An Adaptive Equivalent Circuit Modeling Method for the Eddy Current-Driven Electromechanical System Wei Li1 , Young Woo Jeong2 , and Chang Seop Koh1 College of Electrical and Computer Engineering, Chungbuk National University, Cheongju, Chungbuk 361-763, Korea Electrotechnology R&D Center, LS Industrial systems, Co., Ltd., Cheongju, Chungbuk 361-720, Korea An adaptive equivalent circuit method is developed for analyzing eddy current-driven electromechanical systems in an efficient and accurate way. The problem is analyzed by transferring the system to an equivalent circuit model, which is set up by adaptively dividing the plate into a series of segments based on the field continuity condition at the interface of segments. The performance is obtained by solving the circuit equations combined with motional equations with the Runge–Kutta–Fehlberg method. The circuit parameters, such as self inductance and mutual inductance, are evaluated from the geometry parameters by using an analytic method. The proposed method is applied to analyze two eddy current-driven electromechanical systems, Thomson-coil actuators used for a practical engineering problem and TEAM Workshop Problem 28. The accuracy and efficiency of the proposed method are verified by comparing the calculation result with the FEM calculation result and the experimental result. Index Terms—Eddy current, equivalent circuit method, TEAM workshop problem 28, Thomson-coil actuator.
I. INTRODUCTION
E
DDY current-driven electromechanical system is a kind of electrodynamic repulsion or levitation system utilizing the eddy current induced in conducting plate to generate the electromagnetic force. This is a typical complex system that involves electric circuits, magnetic fields and mechanical movements. This kind of repulsion system is being involved in many engineering applications due to its very fast response via small inductance. Typical examples are actuators of circuit breaker and arc eliminator, electromagnetic launcher, and electromagnetic brake. In order to analyze this kind of system, finite element method (FEM) coupled with circuit and motional equations have been used widely. The FEM, however, require huge computing time related with transient time stepping analysis. Especially if it is related with parameter optimization of the system, it can be hardly adopted. For reducing the computing time, hence, paying attention to the fact that the system does not contain any magnetic material, the problem can be solved by using equivalent circuit method [1]–[3]. In this paper, an adaptive equivalent circuit method is developed for solving the problem, where the conducting plate is divided into a series of segments considering the distribution of the eddy current. The refinement algorithm based on the field continuity condition has been adopted to get minimum required segmentation of the conducting plate for precise performance analysis. The calculation procedure of the proposed method will be introduced through a test model of Thomson-coil actuator which is an eddy current-driven repulsion system used for an arc eliminator in the switchgear system. To verify the calculation accuracy and efficiency of the proposed method, the result is compared with the FEM calculation result and the experiment result of a prototype of Thomson-coil actuator which will be applied in a practical engineering problem. Another typical Manuscript received October 30, 2009; revised January 22, 2010; accepted February 01, 2010. Current version published May 19, 2010. Corresponding author: C. S. Koh (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2042689
Fig. 1. Simplified mechanism of the Thomson-coil actuator.
eddy current-driven electromechanical problem, TEAM Workshop Problem 28, is also used to test the calculation accuracy and efficiency. II. CALCULATION PROCEDURE A. Thomson-Coil Actuator The structure of the Thomson-coil actuator is shown in Fig. 1. The actuator is mainly composed of an exciting coil and a moving plate with high conductivity. Normally, the normal current will flow along the transmission path, as labeled in Fig. 1, and the plate stays at the opening position. At this moment, the fixed contact and the moving contact are separated. If an open arc due to a severe fault in the switchgear is detected by the arc monitoring system, then a trip signal will be sent to close switch S, due to which a pulse current will be generated in the exciting coil. The electromagnetic repulsive force generated between the exciting coil and plate will drive the plate together with the moving contact away to the close position, at which the conducing path of fault current is constructed by the connection of fixed contact and moving contact. Therefore, the fault current will be bypassed to the ground through the conducting path. B. Equivalent Circuit Since the system is axially symmetric, the eddy current induced in the conducting plate will also be axis-symmetric. In order to construct an analytic model which can represent the uneven distribution of eddy current in the conducting plate, the plate is divided into a series of segments, and in each of them, the eddy current is assumed to be uniform. Because the eddy current
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It can be seen that if all the current values are solved and the derivative of mutual inductance are calculated, the force can be obtained easily. The motional equations of the moving plate can be expressed as follows: Fig. 2. Equivalent circuit model of the Thomson-coil actuator. (a) Segmentation of plate, (b) Equivalent circuit.
is axis-symmetric, the individual segment actually corresponds to a conducting ring with circuit parameters of resistance and inductance, as shown in Fig. 2(a). Based on this modeling and considering the mutual inductances between conducting rings, the exciting coil and conducting plate can be transferred, as shown is the self inducin Fig. 2(b), to an equivalent circuit model. is the mutual inductance tance of the exciting coil, while between the exciting coil and the th segment. The circuit equations as well as the flux equations are shown as follows:
(6) (7) (8) where represent the displacement, speed, and are load force and plate mass, recharge, respectively. spectively. The dynamic state equations of the system can be obtained by combining (1)–(3) and (6)–(8), as follows:
(9-a)
(1) (2) (3-a)
(9-b) (9-c) (9-d) (9-e)
(3-b) and are the resistance and flux linkage of the exwhere citing circuit. is the exciting current. and stand for the and are the resistance and flux charge and capacitance. linkage of the th circuit. stands for the eddy current in the th circuit. is the mutual inductance between the exciting circuit and the th circuit. is the mutual inductance between the th and th circuits. As there is no magnetic material in the system, the inductance only depends on the geometry dimension, so in (3), the derivative of mutual inductance to time can be written to the derivative to displacement multiplied by the speed of the plate. The self inductance will not change with the relative position between the exciting coil and the moving plate; therefore, the derivative of the self inductance to the position is zero. Furthermore, all the segments move together as a whole plate, it is unnecessary to calculate the derivative of mutual inductance between two segments. There are several methods that can be used to calculate the self and mutual inductance of coaxial coils, such as FEM magnetostatic solver and analytic methods. In this paper, to keep the continuity of the calculation program, the inductance is calculated by using an analytic method proposed in [4]. and electromagnetic force can be calcuThe energy lated as follows: (4)
(5)
where can be calculated from (5). The dynamic characteristics such as exciting current, force, speed and displacement, can be obtained by solving these ordinary differential equations with initial conditions using the Runge–Kutta–Fehlberg method which can control the time step size automatically. C. Adaptive Segmentation The accuracy of the solution strongly depends on the segmentation of the conducting plate. Finer segmentation of the conducting plate, in general, gives more accurate performance with more computing time. Hence, in order to achieve precise performance analysis with less computing time, an adaptive segment refinement is developed. According to the electromagnetic field theory, the tangential component of electric field intensity should be continuous at the interface of two segments. From this, together with Ohm’s law, . we get, at the interface of segments, the condition of In this paper, local field continuity error for a segment (e), shown in Fig. 3, is defined as follows: (10) where is the number of the neighboring segments of the th and are the eddy current densities of the th segment, segment and the th neighboring segment, respectively, is the length of the overlapped interface between the th segment and the th neighboring segment.
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Fig. 3. Segment refinement. (a) Before refinement; (b) after refinement.
In the first loop, the plate is initially divided into a few segments roughly, after the calculation is finished, the local error for all segments are computed, the segments with big error will be refined into more segments. Therefore, during the next calculation loop the refined segments will be used instead of the old segments. The refinement will not stop until the calculation result is convergent. D. Solving Procedure The overall solving procedure of the proposed method is summarized as follows. Step 1) Divide the conducting plate into several uniform segments, and set up the equivalent circuit. Step 2) Calculate all circuit parameters including the resistance and inductance of all the segments and the exciting coil, and set up the state equations. Step 3) Solve the state equations in time domain, and calculate the local field continuity error for all segments. Step 4) Stop if the final displacement is converged. Otherwise refine the segments and go to Step 2.
Fig. 4. Distribution of eddy current density with different segmentations. (a) The first iteration loop; (b) the third iteration loop; (c) the final iteration loop.
III. VERIFICATION OF RESULT To verify the efficiency and accuracy of the proposed method, two eddy current-driven electromechanical systems excited by capacitor bank and current source respectively are tested. A. Thomson-Coil Actuator A prototype of Thomson-coil actuator is analyzed by using the proposed calculation method. The distributions of eddy current density and refinement corresponding to different calculation loops are shown in Fig. 4. The unit of the eddy current density is Ampere per square meter, and the unit of dimension is in relative value. It can be seen that by adopting the adaptive refinement algorithm, the eddy current distribution in the plate can be represented vividly, and the refinement is concentrated on the part where the eddy current density changes roughly. As the number of segments increases, the distribution of the eddy current density becomes continuous. The prototype is also analyzed by using a commercial 2-D FEM software with a mesh of about 40 000 elements and adaptive time step control. The FEM calculation result is taken as a reference. The comparison of exciting current, force, speed and displacement between the calculation results and experiment results are shown in Fig. 5. The units of all the parameters are in relative value. It can be seen that the calculation result of the proposed method matches very well with the FEM calculation result. However, the calculation results from both the proposed method and FEM show some error compared to the experiment result. This is mainly because the friction that exists in the experiment is not considered in the calculation.
Fig. 5. Comparison of results of the Thomson-coil actuator. (a) Exciting current; (b) electromagnetic force; (c) speed of the plate; (d) displacement of the plate. TABLE I COMPARISON OF CALCULATION ACCURACY AND EFFICIENCY
The calculation accuracy and efficiency are compared between different methods as shown in Table I. If the FEM calculation result is taken as the reference, the relative error of
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Fig. 6. Configuration of the TEAM Workshop Problem 28. Fig. 8. Calculation results of the TEAM Workshop Problem 28 by using the proposed method with different segmentations.
Fig. 7. Distribution of eddy current density along the radius of the plate. Fig. 9. Comparison of results of the TEAM Workshop Problem 28.
the final displacement corresponding to different segmentation can be calculated. As the number of iteration increases, the calculation result becomes more precise. By the FEM calculation, the computing time is about 10 h; while with the proposed method, the computation time is less than 8 min even with 95 segments. Therefore, it is much more efficient by using the proposed method to analyze and design the Thomson-coil actuator. B. TEAM Workshop Problem 28 Fig. 6 shows the configuration of the TEAM Workshop problem 28 where a cylindrical aluminum plate is located above two cylindrical coils [5]. The initial levitation height which is from the upper edge of the current carrying area to the lower edge of the plate is 3.8 mm. When a sinusoidal current source with 20-A magnitude and 50-Hz frequency is supplied to the inner and outer coils in opposite directions, the conducting plate will experience a repulsive force along the direction due to the eddy current induced in the conducting plate. Fig. 7 compares the distribution of the eddy current density along the radius of the plate with different iteration loops. It is observed that the proposed algorithm gives quite smooth eddy current distribution when the adaptive refinement is adopted. Fig. 8 shows the calculation results using proposed method with different segmentation. It can be seen that as the number of segments increases, the calculation result trends to be convergent. The calculation result of proposed method is also compared with the FEM calculation result and the experimental result as shown in Fig. 9. It can be seen that the calculation result from the proposed method matches very well with the FEM calculation result and the experiment result. At the same time it should be noted that the proposed method with 38 segments takes just 2 h, while the FEM calculation requires 89 h.
IV. CONCLUSION An adaptive equivalent circuit method is developed for analyzing the eddy current-driven electromechanical system. The system is transferred to a set of equivalent circuits. The circuit equations as well as motion equations are solved by using the Runge–Kutta–Fehlberg method, through which the numerical calculation error due to time step can be controlled and minimized. By adopting the adaptive refinement algorithm, the minimum number of segments for accurate analysis can be obtained which both improves the calculation efficiency and ensures the calculation accuracy. The proposed method is going to be used to analyze and optimal design the Thomson-coil actuator for a fast response. It can also be applied to other axial symmetrical eddy current-driven electromechanical systems for a fast analysis and design. REFERENCES [1] T. Takeuchi, K. Koyama, and M. Tsukima, “Electromagnetic analysis coupled with motion for high-speed circuit breakers of eddy current repulsion using the tableau approach,” Elect. Eng. Jpn., vol. 152, no. 4, pp. 8–16, 2005. [2] S. M. Lee, S. H. Lee, H. S. Choi, and I. H. Park, “Reduced modeling of eddy current driven electromechanical system using conductor segmentation and circuit parameters extracted by FEA,” IEEE Trans. Magn., vol. 41, no. 5, pp. 1448–1451, May 2005. [3] S. H. Lee, I. G. Kwak, H. S. Choi, S. M. Lee, I. H. Park, and W. K. Moon, “Fast solving technique for mechanical dynamic characteristic in electromagnetic motional system by electromechanical state equation including extracted circuit parameter,” IEEE Trans. Appl. Superconduct., vol. 14, no. 2, pp. 1926–1929, Jun. 2004. [4] T. H. Fawzi et al., “The accurate computation of self and mutual inductances of circular coils,” IEEE Trans. Power Apparatus Syst., vol. 97, no. 2, pp. 464–468, 1978. [5] H. Karl, J. Fetzer, S. Kurz, G. Lehner, and W. M. Rucker, Description of TEAM Workshop Problem 28: An Electrodynamic Levitation Device [Online]. Available: http://www.compumag.co.uk/
